The structure of irreducible matrices †

Let A be an irreducible matrix with index of imprimitivity h is shown that there exists a permutation matrix P such that PAP^t is in a superdiagonal block form with k nonzero blocks if and only if k divides h It is also shown that a matrix in a superdiagonal block form without zero rows or columns is irreducible if and only if the product of the superdiagonal nonzero blocks is irreducible.     

The difference between doubly nonnegative and completely positive matrices

The convex cone of n×n completely positive (CP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative, and it is known that, for n4 only, every DNN matrix is CP. In this paper, we investigate the difference between 5×5 DNN and CP matrices. Defining a bad matrix to be one which is DNN but not CP, we: (i) design a finite procedure to decompose any n×n DNN matrix into the sum of a CP matrix and a bad matrix, which itself cannot be further decomposed; (ii) show that every bad 5×5 DNN matrix is the sum of a CP matrix and a single bad extreme matrix; and (iii) demonstrate how to separate bad extreme matrices from the cone of 5×5 CP matrices.     

An alternating direction algorithm for matrix completion with nonnegative factors

This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are partially known (to a certain accuracy). This problem aggregates two existing problems: (i) nonnegative matrix factorization where all entries of M are given, and (ii) low-rank matrix completion where nonnegativity is not required. By taking the advantages of both nonnegativity and low-rankness, one can generally obtain superior results than those of just using one of the two properties. We propose to solve the non-convex constrained least-squares problem using an algorithm based on the classical alternating direction augmented Lagrangian method. Preliminary convergence properties of the algorithm and numerical simulation results are presented. Compared to a recent algorithm for nonnegative matrix factorization, the proposed algorithm produces factorizations of similar quality using only about half of the matrix entries. On tasks of recovering incomplete grayscale and hyperspectral images, the proposed algorithm yields overall better qualities than those produced by two recent matrix-completion algorithms that do not exploit nonnegativity.     

Separating doubly nonnegative and completely positive matrices

The cone of Completely Positive (CP) matrices can be used to exactly formulate a variety of NP-Hard optimization problems. A tractable relaxation for CP matrices is provided by the cone of Doubly Nonnegative (DNN) matrices; that is, matrices that are both positive semidefinite and componentwise nonnegative. A natural problem in the optimization setting is then to separate a given DNN but non-CP matrix from the cone of CP matrices. We describe two different constructions for such a separation that apply to 5 × 5 matrices that are DNN but non-CP. We also describe a generalization that applies to larger DNN but non-CP matrices having block structure. Computational results illustrate the applicability of these separation procedures to generate improved bounds on difficult problems.     

关于圈的完全正矩阵实现

本文给出了一个关联图为圈的非负、半正定矩阵A为完全正的一个充要条件.我们还证明了这样的矩阵A（当A为完全正时）的分解指数即为A的阶数.     

Possible numbers of positive entries of imprimitive nonnegative matrices

Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n,?k) and m(n,?k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d?≤?M(n,k), there exists an n?×?n irreducible nonnegative matrix A with imprimitivity index k such that?σ?(A)=d.     

Completely positive module maps and completely positive extreme maps

Let be unital -algebras and be the set of all completely positive linear maps of into . In this article we characterize the extreme elements in , for all , and pure elements in in terms of a self-dual Hilbert module structure induced by each in . Let be the subset of consisting of -module maps for a von Neumann algebra . We characterize normal elements in to be extreme. Results here generalize various earlier results by Choi, Paschke and Lin.

     

Matrix completion problems of block type

The solvability, either unconditional or under certain conditions, is proved for three matrix completion problems of block type. Being a problem of block type means that a matrixA must be constructed with a prescribed characteristic polynomial and one or several blocks in a 2 × 2 partition also prescribed. For the problems under analysis, one prescribes, respectively, (a) the blockA ₁₂; (b) the blockA ₁₁; (c) the blocksA ₁₁ andA ₁₂. We discuss our experience in implementing the algorithms that solve the problems above as procedures in the symbolic computation system MAPLE. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 863–873, June, 2000. The first author was supported by the Russian Foundation for Basic Research under grant No. 97-01-00927.     

Cuntz-Pimnser algebras, completely positive maps and Morita equivalence

Let be a completely positive map on and let be the associated GNS--correspondence. We prove a result that implies, in particular, that the Cuntz-Pimsner algebra of , , is strongly Morita equivalent to the Cuntz algebra , where is the index of .

     

正P-矩阵的逆零完成问题(英文)

本文基于不完备P-矩阵的1-通弦图和1-通弦块图的完成问题,获得了不完备正P-矩阵在一定条件下的k-通弦图和k-通弦块图的完成,同时解决了不完备正P-矩阵在k-通弦图和k-通弦块图下的逆零完成问题(k为大于1的整数).     

New results on the cp-rank and related properties of co(mpletely )positive matrices

Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone of completely positive matrices of the same order are dual to each other with respect to the standard scalar product on the space of symmetric matrices. This paper establishes some new relations between orthogonal pairs of such matrices lying on the boundary of either cone. As a consequence, we can establish an improvement on the upper bound of the cp-rank of completely positive matrices of general order and a further improvement for such matrices of order six.     

Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

For a compact subset of symmetric with respect to conjugation and a continuous function, we obtain sharp conditions on and that insure that can be approximated uniformly on by polynomials with nonnegative coefficients. For a real Banach space, a closed but not necessarily normal cone with , and a bounded linear operator with , we use these approximation theorems to investigate when the spectral radius of belongs to its spectrum . A special case of our results is that if is a Hilbert space, is normal and the 1-dimensional Lebesgue measure of is zero, then . However, we also give an example of a normal operator (where is unitary and ) for which and .